 Det er en plads til at være her i dette event, færdigende Marcel Bâché. Jeg havde nogle meget gode interaktier med Bâché, vigtigere ærlig i min karriere. Jeg remælde rigtig vigtigt, når jeg havde et dækken længe før. I hans plej, der var en eksempel med hende og hende. Det var bare hende, hans wife og mig og Sergio Kleinermann. Det var eksempel vigtigt, og han var altid meget vigtig. Så det er meget vigtigt at være her. Jeg vil tage om nogle resultatere om optimale regularitet for geometriske fløjer. Så dette vil være et samarbejde med Bill Minicotti. Hvis en modell er en fysisk fenomen, så er en ofentlig trakking franser, som vil tilføje en fysisk færdigere spørgsmål. Hvis en spørgsmål er virkelig en fysisk færdigere spørgsmål, det løber til en af den klassiske, forventelige og nødvendigste færdigere spørgsmål. Og så vil man se, hvad der er regularitet af solutier til denne spørgsmål. Og så vil jeg tage om det her, at det er ansatte til dette. Så vi kan faktisk, for denne spørgsmål kan vi faktisk prøve optimale resultatere, så vi kan prøve, hvad det er, hvor regularitet af solutier er. Men efter det, kan man fortælle, at vi initiallyer, at kunne solutierne være mere regular? Kunne de egentlig bevægge det? Og så, at jeg vil tage omkring en minut, at vi ved, at solutierne er altid færdigere færdigere. Den større derivativ er altid begyndt. Man kan helt karaktere, når den større derivativ er C2. I generelt er det færdigere færdigere. Men der er eksempel, hvor det ikke er C2. Hvis det er C2, er det i generelt ikke C3, okay? Så vi kan helt karaktere det. Men selvfølgelig, så fortællet med det her, vi er stadig vondt om, selv hvis de har denne lille regularitet, er det stadig muligt, at solutierne kunne bevægge, hvis de er meget mere regular? Og jeg vil tage til dig, hvorfor vi er interesseret i det. Og så er vi... Så de størrelse er sort af det mest optimistiske. Så solutierne, så jeg bare trækker på franskabel, men vi er stadig vondt om at se, om det er muligt, at solutierne kunne bevægge, hvis de var særlig analytisk, okay? Og jeg vil sige, at jeg håber at sige, at det er en case, hvor vi kan faktisk sige det. Og igen, jeg vil tage til at sige om nogle motivationer. Hvorfor tager vi det? Og der er nogle virkelig reason for det. Så jeg vil sige om, at efter, vi talte om det, så efter, vi talte om denne particular, det generer en ekvation, og om at prøve optimistisk regularitet. again, optimistisk regularitet er... i for at prøve optimistisk regularitet, det købe er faktisk at forstå underliggendegeometry. Hvis du var bare tænkt på denne problem, som det har været før, men hvis du var bare tænkt på denne analytisk, så vil du ikke kunne prøve optimistisk regularitet. Du ville bare, som et klassisk resultat, kunne prøve at prøve, at solutierne er lypset. Så... Men med at forstå underliggendegeometry, vi kan prøve optimistisk regularitet resultat. Men nu, efter at tales om denne equation, vil jeg prøve at tage om... Så jeg vil se, om denne teknik sætter til andre equationer, og nu vil jeg skrive en resultat til denne. Så her er det faktisk ideen. Så det er et klassisk ting, at du har en front. I dette stedet er det perimeter af en fire. Det fire er revolveret over tid. Du er tænkt på at trække perimeteret, som det revolverer. Og du har tænkt på et simpelt eksempel, hvor fire initiallyer har to forskellige brænderser over tid, de brænderser samarbejde. Så du kan tænke på, at det er lidt mere kompliceret behavior. Og så her er et annet klassisk eksempel, hvor du trækker en anden front. Det er et olieplæt på vand. Og så i dette stedet kan dropletes komme sammen, og at dropletes kan tænke på to. Så du har den sort af behavior. Du er tænkt på at trække den. Og så det løber til, jeg tager lidt mere om det, men det løber til, eventually løber til det. Det er ligesom en posterboy, eller en posterchild for denne kind af ekvation. Så det er denne, det er denne nonlinear, det generer eliptisk ekvation. Det generer den, fordi det forbereder dig, at du er tænkt på at tænke på normen af graden af funktionen. Og det er løbt at se, at funktionen har altid kritisk punkt. Der er altid punkt, hvor graden vandrer. Så du er tænkt på at tænke på noget, at tænke på 0. Okay, så det er derfor, at denne ekvation generer den. Så det er den eliptisk version af denne ekvation, og der er et stedet på at generere et parbolligt version. Men du ser på den anden side, det er et relativt simpelt ekvation. Det er en vigtig ekvation, det er en nonlinear ekvation, men det er en ekvation, som er modt af den Laplace ekvation. Men det er en nonlinear, og det er en stedet genereret. Så her er det optimale verden, at solutoren til denne ekvation, til denne eliptisk ekvation, er altid samt befandligt. Og så everywhere, at kritikere points. Så det er der, at denne ekvation er, så jeg skulle også sige, at det er, at hvis du er fuldstændig fra, hvor graden vandrer, så er det klassisk, at en solution her er smut. Så hvad du rigtig kender for, er et kritikere point for dig, hvor den ekvation er genereret. Men så den resultat prøver, at det er, at det er kritikere point, den funksjon er stadig træt befandligt, og der er kritikere points, den hæsken faktisk har en meget simpel form. Jeg taler lidt mere om det senere. Og så den klassiske genereret ekvation, det var først sjovt, at du kunne solge denne ekvation i den viskostighed som 25 år senere. Og så det var en af de, jeg taler lidt mere om det også, men det var en af de, en af de, færdige usager af viskostighed, men det turner ud, at funksjonen er træst befandligt, og det turner ud, at faktisk solutions er klassiske solutioner. Du har ikke nødt til viskostighed, fordi de faktisk solver denne ekvation i en klassisk sens. Og igen, når den seneste drivning er existent, og vi kan sige, at det er altid fundet, i generelt er det ikke fortsættet. Så det er en sådan form af optimale regulæritet, prøve at solutionsen er altid træst befandligt. Så optimale regulæritet er obviously en klassisk analytisk problem, men om at prøve denne optimale regulæritet, det løber på at dække, at forstå den underligende geometrige. De ideer, de ideer i prøve, bruger en mix af geometrige, og også af virkelige geometrige. Og så er det alle sammen med Bill Minkatti. Og så, så når man har denne optimale regulæritet resultatet, så further development on this equation that I'll describe, it connects with, so it's part comes from the techniques involved in proving it, but it connects for further development that I'll describe connects with when the terms work on catasor of theory and singularity theory, and we believe that these results, these further development are the first are the first instances of a general principle, and this general principle is the solutions of many degenerate equations behave as if they are analytic, even though they have much lower regularity. If this general principle is true, then it would actually explain various conjecture phenomena. So I believe that these, I believe if they turn out to be true that this is a general principle, I believe this will have, I mean it will have various applications to kind of well-known phenomena that really current techniques have no way of addressing, okay? And so let me come back to the equations, I'll describe all of these things, but I want to come back to the basics first. So I want to come back to this elliptic degenerate equation, where does it come from? So it comes from that you're looking at a hypersurface, the hypersurface will eventually be thought of as this moving front, but at the moment I'm just looking at a fixed hypersurface, and it's the unit normal of this hypersurface. H is the mean curvature scalar, so this is just the divergence of this unit vector field. And so if you want to write it out, so the divergence is just a sum of the derivatives, and then in a product with the corresponding EI, where EI is this orthonormal basis for the tangent space of the hypersurface at a given point. So this is the mean curvature at that given point. Now, so this is just, this is what the mean curvature is, this is what the unit normal is. Now suppose that the hypersurface is a level set of a function. So suppose that you have some function on your cleaning space, suppose S is a regular value, and the hypersurface is a preimage of S. Well then obviously the unit normal is just a normalized gradient, and therefore the mean curvature is just given like this. So the mean curvature flow is then one parameter family of hypersurface, where the point at each at a given time, the point is evolving by it's moving orthogonal to the hypersurface, and the speed is minus the mean curvature, right? So if you have a one parameter family of hypersurface that is moving by this equation, then it's set to flow by the mean curvature flow. So this is a geometric heat equation. It was first started by Birkhoff, the first reference I know of it, but there might have been earlier, is Birkhoff in 19th teens. Independently it was started in the material science literature already in the 1920s. If you were a physicist you would think of the mean curvature as the surface tension, and the equation has this property that the area decreases the most efficiently. Okay now so here's the simplest examples of mean curvature flow. If we're looking at it in our tree, if we're looking at a one parameter family of round two spheres with radius, so here I'm thinking about time as being negative, and I'm looking at a one parameter family of round two sphere of radius square root of minus 4t all centered at the origin. This one parameter family of two spheres flows by the mean curvature flow. If you're looking at cylinders, so round cylinders where the radius, so this again I'm looking at our tree, I could look at or similarly I could look in higher dimension, but if I'm looking at round cylinders where the cross section, so this is a circle round circle where this have radius square root of minus 2t, t is again negative, then you will see that these here actually flow by the mean curvature flow, and you observe that at time t equal to zero, the surfaces are not any more surfaces, like the first one here have shrunk to a point, and the second family here has shrunk to a line, so this is correspond to that the evolution become extinct at that time. Right after that time it disappeared, it became singular at that time, and right afterwards it's gone. Okay, and here you see these examples, and the even more trivial one, but less interesting is just the plane, where the mean curvature is zero, so nothing happened, it's static under the flow. But these spheres and the cylinders and also the plane, they're self similar under the flow, this means that the shape at a later time is exactly like at an earlier exact it's scaled, in this case it's scaled down. So they're self similar in this way, and so two very simple but key properties are, so one is that the mean curvature, and this is part of why the mean curvature flow is interesting, it's a gradient flow for area, so under the mean curvature flow the volume or area decreases the most efficiently, and the second basic property is what's called the avoidance principle, the avoidance principle is just a geometric formulation of the maximal principle, and so they say that if you have two initially disjoint surfaces, so these surfaces are just illustrated as if they were curved, so the red one is one surface, the blue one is another surface, if you have two surfaces they are initially disjoint under the flow, they will remain disjoint, and again this is just the maximal principle, and the idea is that if they weren't, if they didn't remain disjoint under the flow, then there would be some first time where the red one, the outer one would have just caught up with the blue one, so since it was the first time, this would mean that the blue one would have to lie in, still would have to lie inside the red one, but would touch at just one point, but at that point you see that the mean curvature of the red one is actually smaller than the mean curvature of the blue one, which means that the blue one is moving faster, so it means that if you go back a little bit in time, the blue one must have crossed the red one, so that's the avoidance principle, and this is really another way of thinking about it, like this is just sort of a geometric term, it just, if you write down the equation, it just boils down to the maximal principle, but in one dimension the mean curvature flow is called the curve shortening flow, and the simplest example is that of a round circle, so under the flow it shrinks through round circle, and in finer time become extinct in a point, and then there are a more interesting example that are shown next, which is sometimes referred to as a snake, and so let's just see this, so this first example here, it's illustrate a theorem of Gage and Hamilton from the early 80's, and so this theorem of Gage and Hamilton say that if you take a convex curve, so the outer one here you should think of as the initial curve, so if you take a convex curve in the plane, and you run the curve shortening flow, the convex curve under the flow, so again so the outer one think about that as the initial curve, the first one inside the outer one think about that at some slightly later time, and then all the other one you should think of as later and later times, and so the theorem of Gage and Hamilton say that if you take a convex curve under the flow it will move inside itself, and it will eventually at some time become almost round, and then once it become almost round it will keep shrinking and eventually disappear in a point, and that point if you kind of rescale right before it disappear, it would look like almost perfectly round circle, okay, so this is a theorem of Gage and Hamilton from I believe 84, and so now Gage and Prude it may be 82 or something like that, Gage and Prude in 89 he proved that if you take any, so I should also have said that my surfaces and in this case my curves are always, my surfaces are always embedded, if it is a curve then we usually, we don't talk about a curve as embedded, we usually say that it's simple, and typically my surfaces and the same curves are closed, so there's a theorem of Gration that say that if you take any simple closed curve in the plane, so this here is supposed to be a simple closed curve, that is kind of wrapping around itself a number of times, if you take any simple closed curve in the plane and you run the curve shorting flow, so this here is the initial time, this is at a slightly later time, slightly later again, etc. down here, if you run the curve shorting flow then actually in this case if you take any simple closed curve it will start unwinding, it would unwine, it's going to shrink, and the reason why it will shrink is comes from the avoidance principle, because if you take any simple closed curve you can just encircle it by a large round circle, you know that the round circle in finer time will become extinct, so and whatever is inside it will remain inside it, so that will also disappear, so it's going to shrink, might any simple closed curve is going to shrink, but the theorem of Gration say that it's actually unwinding faster than is shrinking, and so this means that what happens by the theorem of Gration is that it will unwind and eventually at some time it will be quite small, but at that time it will be convex and then the theorem of Gage and Hamilton take over and they say that once it becomes convex then as time goes by it becomes rounder and rounder and it will eventually disappear in a point, so this is the theorem of math Gration from 89, any simple closed curve in the plane shrinks to a round point in finer time, okay so that's kind of the story, that's conclude the story for the curve shorting flow in so for the curve shorting flow, but in higher dimension various interesting phenomena happen, so here's a very simple example of a surface of evolution, this is a torus of evolution, if you're looking at this as the initial surface, you run the flow, the mean curve of the flow will preserve the symmetry, so it will remain a surface of evolution, and it will shrink, the cross section here will shrink and it will disappear in finer time, it will disappear in a circle, okay, there's another example, more interesting example of math Gration, also from 89 and they say that if you take a dumbbell, so take a dumbbell, so this is a dumbbell in R3, so it's rotational symmetric around an axis here, and in this case the two bells here have equal size, there's also actually in this particular example there's also a reflection symmetry around here, so this example of Gration say that if you take a dumbbell like this connected with a sin bar here, this sin neck, then under the mean curve of the flow the neck here will pinch off first, cutting the surface into two pieces and later each of these two bells will shrink to a round point, so this is illustrated, this is actually a computer simulation done by a guy called Uwe Meyer, and so, and he also did, he was also so nice to do the computer simulation for the theorem of math Gration earlier, so here again, so this is the initial time, this is a slightly later time, right here you see the neck have just pinched off, the surface becomes, it's a heat equation, so the surface actually becomes, it doesn't really look like it here, but it becomes instantaneously smooth after the neck have cut off, and it will evolve, keep evolving and it will smoothen out here, and the two bells become actually convex and then eventually disappear in a point, each of them, and so this here, this example here, so Gration had a kind of crude way using the avoidance principle to say that the neck will really break off first and break into, the surface will break into two pieces, slightly later in 91, this example here sets inside a much larger family of examples of surfaces that are rotation symmetric, so all surfaces that are rotation symmetric around an axis like this was completely analysed in around 91 by Altshuler, Ankenen and Giger, and also some of the, they analysed sort of the family in complete generality of surfaces that are rotation symmetric with some further assumption, similar results was proven at the same time by Sonar and Sognidis, so again, so they analyse completely if you have a surface, initial surface, that is rotation or symmetric around an axis here, they understood completely what happened, and the basic method that they were able to apply, which is of course very particular to that example, but still very lovely, is a stormy real comparison theorem for that exists, but it really only exists for a one dimensional heat equation, okay, so there's no other version of that, and so, and that was a theorem, this stormy real comparison theorem was proven earlier, I mean of course there's a very classical one, but that's not for the heat equation, the one they're using is one for the heat equation, that was proved earlier by Ankenen's, okay, so that was the basic technique, and right, and so, but you see also in this sort of picture here, it begs, or already here it begs to make sense of what it means to flow in the case where it breaks into pieces, there is a moment where the surface is not any more smooth, and yet you are continuing the flow through that point, so one of the next slides are going to be talk about how to make sense of the flow past a singularity, and this was this was work of Chen, Giga and Gouda, and Evansen's book, and I'll get to just a single, and so this is really what's called the level set method, the level set methods have been extremely successful numerically, and this was numerically it was brought into play by Osha and Setschen in 89, and they have a method for they're looking at these equations where you're tracking a moving front, and again as I said in the beginning, the prime example of this is where the speed is a curvature, so the one we're looking at, and they develop something called the level set method, it was earlier considered, this idea was earlier considered certainly in the material science literature, was considered but they you know they weren't able to do anything with it, but they in the material science literature people already understood, how to track you know that there's a reasonable notion of tracking as surface past a singularity, but they couldn't prove anything about it, but they just wrote down what is now known as the level set equation, okay, so the basic idea here of Osha and Setschen is that you have this moving front and you want to track what happened to this front as it's moving with this curvature dependent speed, and so you choose a function, so you want to you want to take the geometry away from the equation, right, so instead of looking at the surface as it evolved, you want to reduce it to a question about PD, about some function satisfying some differential equation, right, and so the idea is that you take, you choose an initial function, so you have this initial hypersurface, all our hypersurface are closed and embedded, you take a function that has this hypersurface as a level set, and let's say it's positive in the, so you have this hypersurface here, and let's say that you take a function that is positive in inside, negative in on the outside, and let's say zero on the hypersurface, and now the idea is that you want to flow, you don't just flow this hypersurface that is given to you, you want to flow all the level sets, so not just the zero level set, but all of the level sets, and the simple idea is that if you flow all of them, then you get an equation, differential equation in the function, right, and so now the geometry is gone, it's a kind of, it seems like a simpler question, where you're reducing it more classically to a PD question, okay, and so by the avoidance principle, right, if you're flowing, if you're taking two different level sets, well we know that those level sets will remain disjoint under the flow, okay, so again, so this leads to this degenerate parabolic equation, these have been tremendously successful, numerically I gave a talk about this at Berkeley maybe a year and a half ago, since then there have been more than 2,000 references to this paper of ocean and sejtion, so it's extremely, it's extremely used all over the place, but so ocean sejtion, they, in their celebrated work, they were able to give a numerical scheme to, that would track solutions to this equation, not much later, a couple of years later, in the work of Avensens book, Tjenke i Gangutto, they were able to provide the analytical foundation for this equation, and so they were able to prove that you had continuous solutions, the solution, so it's a differential equation, it's a second order differential equation, and so they were able to prove that actually you had solution in this weak sense, which is the viscustic sense, and that solutions were unique, and that they agree with classical solutions, right, I should say I always say this because of my, somehow because of my childhood, I always feel obligated at this point to pause for a moment and say that there was, so the, so the, this notion of viscosity was in the early 80s, and it was developed by Crandall, Mike Crandall and Pierre-Louis Lyons, this result here again of Avensens book, Tjenke i Gangutto was really one of the major success in the theory of this notion of weak solution, but there was an earlier version of viscustic solution, there, that I'm sure everybody here know, but I still feel obligated to mention it, of Calabi from the 50s, and so their basic idea is exactly the same, but the two notions are a little bit different though, and so they are a little bit different, the basic idea is exactly the same, Crandall and Lyons was unaware of Calabi's work, Calabi had envisioned, actually, there's no doubt if one, it's not long ago I read Calabi's old paper, and there's no doubt that Calabi, he, as I'm sure everybody here knows, his immediate applications was a geometric application, but there's no doubt that he had envisioned that it would be useful in purely analytical problems like this, okay, but it's different, it is different though, it's a little bit different, although the basic idea is the same, so now let's look at where the initial hyperservice is mean convex, so in this case Avensens book, Tjenke i Gangutto by by work of Avensens book Tjenke i Gangutto and Brian White, mean convexity here is preserved under the flow, the hyperservice will move inward and it will keep moving inward, because it keeps being mean convex, it will sweep out the entire domain that the initial hyperservice bound, so you are at a picture that looks, let me just get back to this picture here, so if you imagine that this, you have a similar picture, although you don't have this conclusion in here, but if this initial surface is mean convex, then it will move inwards and it will keep moving inward into itself, it will sweep out the entire domain, but you won't have this conclusion here, it does not hold, if it's just mean convex, in fact all of the examples I looked at så far, sorry, this here is actually a mean convex example, and this here is also a mean convex example, if you do it properly, it's not all dumbbells that are mean convex, but you can, if the neck is thin enough and you construct it right, it will be mean convex, okay, so okay, and so now this leads to what's called the arrival time, so the arrival time is, so this is just that, so if you take a general hyperservice and you use this method of oceanization, so this level set method, then you get, then you, then it leads to a function that's satisfying a degenerate parabolic equation, if the initial hyperservice then is mean convex, then this degenerate parabolic equation is actually equivalent to a degenerate elliptic equation, and that's the one I looked at, and I will focus my attention on that, on the elliptic one, though many of the shots apply more generally, okay, but I'll focus my discussion on the elliptic equation, and so this is what's called the arrival time, so remember that our initial surface here is supposed to be mean convex, so at a slightly later time it will have moved inside itself, it will keep moving inward, and so this picture here, which is supposed to be mean convex, and please don't look at it too closely, this was actually produced by Sethian, but it doesn't look very mean convex up there, but anyway just ignore that little error, then this is of course the dumbbell, and it's like you just slice the dumbbell, and so what you see here are the hyperservice at different times, the arrival time is simply just that you take this domain that the initial hyperservice bound, and you're looking at some point in the domain, and you're looking at at what time does the hyperservice come through that point, that's the arrival time, okay, and the arrival time satisfying this equation that I wrote initially, so I should speed up a little bit to see, and so in the case of for the round spheres, shrinking round spheres in R3, the arrival time would be, this function is quadratic polynomial, this corresponds to that, this is just normalized so that it arrived at the origin at time zero, similarly for the cylinders the arrival time function is this quadratic polynomial again normalized so that they arrive at this singular line at time t equal to zero, and so you see two things here, you see that the singular point, so in this case here the singular point, so the only the singular point is at the origin, but that is exactly where this quadratic polynomial has a critical point, and similarly you see that in this case here the singular for the flow, the singular singularities is just the line, but that's exactly the critical set of this quadratic polynomial, and so you can actually prove this more generally that critical points for the function are exactly singularities for the flow, so if you want to think about it geometrically you think about the singularities for the flow, if you want to think about it analytically you think about the critical point for the function, and the second thing is this, but it's these examples are too difficult to fully appreciate this more general principle that somehow you should think about this function as part of a tail expansion of this, but you don't see any other term so it's maybe a little bit hard to appreciate this point, but this will be an important point later, okay, and so here's this equation, this is the arrival time equation, and these I should also say that, so this is, so here we're looking at where the front is moving monotromely, numerically these have been numerically modeled even more efficiently, this case here has been modeled more efficiently, in particular there's something called the fastmouthing method of session that is dealing with this case where the front keeps moving inside itself, okay, and so this is the equation, and what you do is, so this is what Evanston spoke that and independently at the same time Chen, Giger and Gutter, they revert this equation in this form here, right, and it's when you rewrite it in this form here, then it lends itself to talk about that you have a solution in a viscosity sense, it really comes from dividing it into this form here, okay, okay, Tom Ilmen and theses proved that, published a few years later, but he proved that there are solutions to this equation here that are not C2, so in general solutions are not C2, and Tom's example really came actually from the exact dumbbell that we looked at earlier, it came from a particular dumbbell, sorry, like this that has two belts of evil size, it had a reflection symmetry like this, and you're looking at the rival time there, if you constructed the exactly right, you know, with the exact right symmetry, he was able to prove that the arrival time function is not C2, okay, and so, right, and so it turned out that that's theorem of Tom actually is, so Bill and I proved that, so remember that we proved earlier that this function is always twice differentiable, and at critical point the Hessian is in a particular nice form, the second derivative is always bounded, but we also proved that, so this is this theorem here, uh, and, uh, right, and so let me just go to this, and come back to this thing from a previous slide just because I kind of got a little bit ahead of myself, that there's also a converse to this, and they say that when the function is C2, so that you can completely characterize when the function is C2, right, so in general the function is always twice differentiable, the second derivative always bounded, but when the second derivative is actually continuous you can completely characterize it, and so the second derivative is continuous if and only if the function has only one critical value which corresponds to that the flow only becomes singular at one time, there's only one time where the flow is singular, and moreover the critical set, or if you want to sing it geometrically, the singular set for the flow is either a single point where the Hessian is exactly like the Hessian of the arrival time function in the spherical case, or it's a simple closed curve just like it was in the case of the donut, the rotational symmetric donut, and in that case the critical at the single set was just this round circle, so you can prove that the arrival time function is C2 if and only if you either have one of these two cases, there's a single point where a single critical set, sorry, there's only one critical value and there's only a single point where you have, where that is critical for the function, or there's a simple closed C1 curve that is the critical set for, that is exactly the critical set for this function, and in that case the Hessian is exactly like a cylinder, and in the second case here the kernel of the Hessian is exactly tangent to this singular curve, and so you see that these two here are illustrated in these examples, if you take the sphere then you would be in, then the arrival time function is, well it was actually a quadratic polynomial, so certainly C2, and this exactly falls into the first category, the cylinder of course falls in exactly to this case here, the ring is also C2 and this falls exactly into this second category, the dumbbell, well there are two different singular times, there's one time where the neck broke off, and then there's a second time where each of the two bells disappeared, so it doesn't, there's not just a single critical value, so it doesn't fall into any of these category, so it's actually, it's not C2, but that's in the particular case where it had this reflection symmetry, this recover of course this theorem of Tom, but of course this here I'm nothing to do with any, I mean this is just in complete generality, we're not assuming any kind of symmetry here, I should also say that I'm stating the theorem as if it was an R3, but it actually holds an all dimension with you know the kind of trivial modification, so, but it's just for simplicity, I just stated in R3. Okay, so again we have talked about you know this singular for the flow, why this when the hypersurface are not smooth, singular points for the flow, analytically you would think about them as critical points for the arrival time function, here were these two examples, I'll just go a little bit faster because I also think I won't get to the end, and so now here it's just very briefly, here's very briefly the key points in the proof, so if you want to prove that a function is twice differentiable, well of course you know you want to prove that it's a quadratic polynomial up to higher order term, but if you want to prove that in a point the function is twice differentiable, then you want to prove that there's a kind of tail expansion, so there's a quadratic polynomial plus higher order term that approximates the function in the neighborhood of that point, now of course again as we talked about in the beginning we're looking at this degenerate equation, but it's only degenerate at critical point, so away from critical point by classical result you know that solutions are smooth, so you only need to check this that is twice differentiable at critical point, the second order approximation here, this quadratic polynomial is going to be the arrival time, we want to prove that it's the arrival time of the cylinders, so we have a great candidate for this approximation, so this suggests that the level sets if you're looking at you have this critical point where you want to prove that the function is twice differentiable, so you want to approximate the function in a neighborhood of that point by this quadratic polynomial, which is this arrival time function for the shrinking cylinders, then it suggests that near that point the level set of the original function looks like shrinking cylinders, and that is indeed true, and this was already known before, now what the key point though as you may recall if you are trying to prove that even a function is just one time differentiable, to say that a function is differentiable means that it is up to a higher order term, a linear function on all small scales, on all sufficiently small scales, it looks like a linear function up to higher order terms, on all small scales, and that linear function is the same on all scales, it doesn't do you any good if it looks like one linear function on one scale, then you go to a much smaller scale, it looks like a different linear function, and it's the same sort of thing here that is a key to proving that a function is twice differentiable, and this was really the difficult part here, so this here is illustrating, and this here was proven earlier by Bill and myself, and so this is supposed to illustrate that, so this is, so in the middle of this say yellow cylinder right there inside it, think about that as the critical point, and you want to prove that you would like to prove that a function is twice differentiable at that point, and so imagine that the level set of this function right before, at a time right before, the level set looks like in a neighborhood it looks like the yellow set, and look, but imagine also that at a slightly later time the level set looks like the blue set, and even later time it looks like a red set, if this here was possible, then the function wouldn't be twice differentiable, and so Bill and I proved earlier that this here is not possible, that it actually has, that these cylinders will have a limit as you're going down, right, if you, okay, and so this is uniqueness, and what made this really complicated, this case here and required new techniques is that these level sets here are, it basically comes from that the cylinders are not compact, okay, so right, so that's the key, the key to proving twice differentiable is proving this kind of uniqueness, okay, now, so now I want to get into, I want to get into some more recent stuff, some very recent stuff, and so after we know the optimal regularity of this equation, so we're not done actually, because we wonder our initially, initially, actually initially I woke up, you know, few years ago, Christmas morning, and I was just wondering what if solutions, that was before we've proven all of this, what if solutions were always analytic, okay, why couldn't you conclude, and so actually you can, you can, if it was analytic, you can actually solve a bunch of conjectures that are well known, now of course, now we know that the function is not even C2, okay, but you still wonder, okay, it's not analytic, maybe, but what if it's, but it's not, you know, you don't a priori need the function to be analytic, you just need it to have the property that analytic function has, okay, and so this is where it connects with this work of Renate Thomas, I guess Renate Thomas here many years ago, and his work in singularity theory and care theory, a catastrophic theory, and so, so, but the background is that if you take a analytic elliptic PD, then solutions are analytic, again degenerate equation has much lower regularity, and solutions only exist in viscosity sense, now if we, so now we are interested, and again we think that these are the first interest of this general principle, that for many degenerate equations, solutions, they won't be very regular, but they would behave as if they were even analytic, okay, and so this is, this is the first result that we proved Bill 9, we proved that in the case where the function, remember that if the function is C2, the function won't be C2 in general, but even when it's C2, it's in general not C3, it's, you know, it's like, it's like there's almost no functions where C3, even when it's C2, okay, so okay, which remind me by the way, I should have just, I should have skipped a slide and I forgot to come back to it, that was kind of relevant for this discussion, this was this here, I should have said that in the convex case, in the convex case, Frisken proved in 1990, that the arrival time function is always C2, Bob Cohen and Silver Saffati proved in 2006, that in the plane, if you take a convex curve in the plane, then the arrival time function is C3, and Natasha proved, and Natasha Sesson proved in 2008, that if, so this is in the convex case, these are always solved where the initial surfaces convex, that in general the arrival time function is not C3, in fact her result is a little bit better, I mean it proved that essentially, it's even when it's C2, which it is in this case by Frisken's, it's for even when the initial surfaces convex, if you perturb it in a kind of arbitrary direction, it's not going to be C3, okay, so now okay, so now coming back to this, and so I'm skipping this, I mean the idea of this here is using, you know, it's coming, some basic ideas coming here to prove this uniqueness comes from, we got that break geometry, the key point here was though it required entirely new ideas, and this comes from the non-compactness of the cylinders, okay, and but the first thing Bill and I proved was, you know, to go a little bit further, that when the arrival time function is C2, well then we proved I wear a shape it's inequality, so this here is a wear a shape it's inequality, and we conjecture that you have, even when the function is not C2, you have a similar wear shape it's inequality, it's known that you can't get this power here, this power won't hold, but you only need the wear shape it's only asking for a power greater than one, okay, so now we proved this, this was not too hard actually, you know, once we have all the other stuff with uniqueness and all of the other stuff, so this here was not too hard, this was really not our goal here, but what we were interested in further properties of analytical functions, now I should say though that the wear shape it's, so this is a really basic inequality altogether, but it's certainly a very celebrated inequality in real life break geometry, and so a basic consequence of once you have a function and you have that inequality, then you will immediately get what's now known as wear shape it's theorem, and the wear shape it's theorem say that if you have an analytic function, so wear shape it's proved this inequality, not with this power here, but with a power strictly greater than one, with a constant C, both C and that power allowed to depend on the function, okay, and so he proved this for all analytic function, and he used it to prove, he used this kind of, this is the wear shape it's gradient inequality, there's another wear shape it's, and they are actually closely connected, they are actually kind of equivalent, and there's also another wear shape it's inequality, and he wear shape it's used these, this is in the late fifties, early sixties, he used it first to prove a conjecture on swads, which was called the swads division conjecture, which was about distribution, if you have a distribution can you divide it by an analytic function, so use this inequality to settle that conjecture, Hermander at the same time also proved that conjecture, except Hermander couldn't actually quite prove the conjecture, he could only prove this kind of inequality for polynomiums, and so that's why Hermander's name is not attached to this because he only established independently at the same time, but only a special case of it, and wear shape it's went on, and he used it to a few years later, he used it to prove another kind of well-known conjecture at the time, also in real algebraic geometry, which was called witness conjecture, and it was about deformation re-contract of analytic sets, okay, so this is a kind of famous inequality, and okay, and it leads immediately, and this is kind of the basic point in the, in wear shape it's prove of the witness conjecture, it leads immediately, once you have an inequality like that, it leads immediately to this wear shape it's theorem, and they say that for analytic function, if you take a gradient flow line, and that gradient flow line, so you flow, the gradient flow line has a limit point, then it has a unique limit, in fact body prove this, that this gradient flow line has finite length, and hence it has a unique limit, okay, so now Renner-Thom just to come back to this place, and Renner-Thom, ja sorry, ja 5 minutter, ja, so Renner-Thom was, so in the late 60s, early 70s, Renner-Thom was interested in singularity theory and catastrophe theory, which is all about gradient flow for analytic functions, okay, and so Renner-Thom was interested in a generalization of wear shape it's a theorem, and so what Renner-Thom conjectured was, that if you take a gradient flow line, so imagine that this here is a gradient flow line of analytical function, so if you take a gradient flow line of analytic function, well by the wear shape it's theorem, we know that if it has a limit point, it has a unique limit, so it kind of will go towards that limit, but Renner-Thom went one step further, and he said why not, why shouldn't the secant, so the secant is that you take this point here, you're looking at this point, as you're going in here, you're looking at the line segment that goes from the limit point to the point where you are, so this give you a line segment, and Renner-Thom was saying that if you renormalize that line segment to unicize, that's what's called the secant, and Renner-Thom conjectured that the secant would have a limit, okay, and so this means that this figure here, which is some sort of Archimedean or logarithmic spiral would be impossible, this here is supposed to, the red line is supposed to illustrate the critical set that the curve, it is a three dimensional picture, and so you have this critical set and you have the spiral into that, okay, so that was Renner-Thom's gradient conjecture, there was a lot of work on that, the Renner-Thom's gradient conjecture was finally settled in 2000 in a paper by, in a paper, I don't want to remember, there's three Polish guys in a paper of annals in 2000, Bill and I, but actually there's a stronger conjecture, so Renner-Thom maybe here, but I guess I was too early, but somewhere in Paris Renner-Thom was talking about his idea about singularity theory and catastrophe here, and Arnold was in the audience, and Arnold went at one step further, and that one is still unknown for analytic function, and so the Arnold's conjecture, which is a refinement of Renner-Thom's conjecture, said that the velocity, the unit velocity, as you are going towards this critical point on a gradient flow line, then the velocity of the unit velocity should converge, okay, this immediately implies Renner-Thom's gradient conjecture, that's unknown, that remain unknown for analytical function, but in this case of these degenerate functions, Bill and I can prove this, so we can prove both, I mean even the stronger conjecture of Arnold's, and so in the, and again I want to emphasize that there are, that if this principle is true, that these solutions to these degenerate equations behave like analytical functions, it will solve various conjectures, known conjectures for flow, I just mentioned one immediate one, but there's various other, that it is believed that a flow should only go through finally many singular terms, and this would be, there's no, as far as I know there's no way of approaching this, but this gives you a way of approaching this, so then in the last minute I have, I want to talk about other equations, okay, and so I want to talk about systems of equations, so a particular system is the richer flow, so there the metric is evolving by, so over time the metric is evolving in the direction of the richer curvature like this, shrinking cylinders are the singularity models, by result of these guys just because I'm running out of time, I won't mention them, but they are there in two and three dimensions, shrinking sorotons are spheres and cylinders in higher dimension, there are many more examples, it's expected that in all dimension the prevailing singularities are cylinders, here I would prevailing, I mean in terms of dimension reduction, so these are supposed to be the typical singularities, the second most typical singularities are expected to be also cylinders like that, and so Bill and I can prove that for the richer flow cylinders are isolated as shrinkers, and we believe we also can prove the corresponding uniqueness that if you have a tangent flow that is a cylinder, then all other tangent flows are that cylinders as well, and so this theorem here is related to earlier results that we did, and this is isolated in a very strong sense, which is a sense that is relevant for this problem, and it just says that if you have any other shrinkers that is close to one of these cylinders, it's a shrinker for the richer flow, if it's close to a cylinder like this on a large but compact set, you only need to know a compact piece of it, then it must be that cylinder, so I'll stop here, it just could have all the time. In the dumbbell example, how are you moving past the singularity, are you doing surgery? No, no, so I mean that's where the level set method come in, because the level set methods, it gives you this degenerate equation, and the work of, so that's exactly what the work of Evans and Spook and Chen-Gi-Giangotto is, that you can make sense of solutions even at critical point for this equation, so that's, so that, I mean again, so there are these, you know for, you know, there are these different notions of going through singularities by the sort of class, I mean there's at least three classical, there's one of the Georgi, there's one of Brake, you know, using geometric methods, the Georgi using regularization, and then there's the level set method.